Dipole in a Uniform External Field: Torque, Frequency, and Time Period

Dipole in a Uniform External Field is in stable equilibrium and is in the direction of the electric field vector and is parallel \(\overrightarrow{P} ||\overrightarrow{E}\). An electric dipole is a pair of electric charges possessing equal magnitude but opposite charges separated by distance.

• An electric dipole is a separation between positive and negative charges of equal magnitudes.
• Dipole in a uniform electric field experiences torque which is a force in rotational motion or an angular force.
• If a dipole is placed in a uniform electric field, the torque will make it rotate.
• The torque aligns the dipole parallel to the direction of the electric field with zero net force.

Read More: Difference between Torque and Moment

Key Terms: Torque, Dipole, Electric Dipole, Magnetic Dipole, Calculating Torque, Angular frequency, Oscillations, Time period, Electric field, Angular force, Angular velocity

Torque and Its Calculation

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Let us assume that the dipole is introduced to a uniform external field in order to seek out torque experienced by a dipole when placed in an external field. This field will provide an electrical force having qE magnitude in the upward direction for the positive charge, while downward direction for the negative charge.

• We can locate that the dipole rests in translational equilibrium as the net force is zero.
• But what's rotational equilibrium? Taking this case into consideration, it is possible that the dipole might remain in a stationary position but does rotate with a specific angular velocity.
• This fact has been experimentally proven and unveils that both electrostatic forces (qE) work as a torque being applied within the clockwise direction.
• Hence, when placed in the uniform external field, the dipole does get to rotate. 
• It is worth noting that torque always operates in a couple.
• Additionally, its magnitude is equal to the resultant product of force and its arm.
• Here, the arm can be considered as the distance between the point at which rotation takes place for the dipole.  

Dipole Placed within the Uniform External field

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The origin is considered the point when a dipole is placed in the uniform external field. Moreover, torque is represented by the symbol ‘τ’. It must be remembered that torque is a vector quantity. As per mathematical basis:

Torque magnitude (τ) = q E × d sin θ

⇒ τ = qEd sin θ

But qd = p (dipole moment)

⇒ τ = pE sin θ

Therefore, we will say that the dipole moment and cross product of the electric field refers to the vector form of torque. 

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Derivation of Torque of an Electric Dipole

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Derivation of Torque on an Electric Dipole can be understood from the figure given below.

Consider a dipole with charges +q and –q forming a dipole since they are a distance d away from each other. Let it be placed in a uniform electric field of strength E such that the axis of the dipole forms an angle θ with the electric field. Refer to the following image for the derivation.

And the direction of the dipole moment is from the positive to the negative charge; it can be seen from the above equation that the torque is the cross product of the dipole moment and electric field. Notice that the torque is in the clockwise direction (hence negative) in the above figure if the direction of the Electric field is positive.

Oscillation of an Electric Dipole in a Uniform Electric Field

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Consider an electric dipole of dipole moment p placed in uniform electric field E at an angle θ. The torque experienced by the dipole is given by

τ = – pE sinθ

If θ is very small then, sin θ ≈ θ

⇒ τ = – pEθ

Let I will be the moment of inertia of the dipole and α be angular acceleration, then

τ = Iα

From the above two equations, we get

Iα = – pEθ

⇒ α = – (pE/I)θ

Comparing the equation with the equation of simple harmonic motion α = -ω²θ, we get

ω² = pE/I

Also, ω = 2π/T

⇒ 2π/T = pE/I

\(\Rightarrow \omega = \sqrt {\frac {pE}{I}}\)

But, ω = 2π/T (T is the time period of oscillation of the dipole)

\(\Rightarrow \frac {2\pi} {T} = \sqrt {\frac {pE}{I}}\)

⇒ \( T = 2\pi \sqrt {\frac {I}{pE}}\)

Also, T = 1/f (Where f is the frequency of the oscillation)

⇒ \( f =\frac {1}{T}= \frac {1}{2\pi} \sqrt {\frac {pE}{I}}\)

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Previous Year Questions

1. In finding the electric field using Gauss law the formula….[JEE Main 2020]
2. two charges of equal amount +Q are placed on a line...[WEBJEE 2016]
3. A thin disc of radius b=2a  has a concentric hole of….[JEE Main 2015]
4. The average velocity and the average speed of the toy car between 0to30to3 seconds are respectively….[NEET 2018]
5. The electric flux linked to the surface, in units of volt m is...[NEET 2010]
6. maximum torque exerted by the field on the dipole is….[KEAM]
7. What happens when some charge is placed on a soap bubble?
8. On charging by conduction, mass of a body may​…
9. Object may acquire an excess or deficiency of charge by….​
10. In figure, two positive charges, q2 and q3 fixed along the yy axis, exert a net electric force in the +x direction on a charge q1 fixed along the x axis. If a positive charge Q is added at (x, 0), the force on q1​
11. In a medium of dielectric constant K, the electric field is →E→ . If ∈0 is permittivity of the free space, the electric displacement vector is….​
12. If an electron has an initial velocity in a direction different from that of an electric field, the path of the electron is…..
13. If a body is positively charged, then it has….​.
14. If a body is charged by rubbing it, its weight….​..
15. If 10⁹ electrons move out of a body to another body every second, then the time required to get a total charge of 1C on the other body is
16. Four point charges are placed at the corners of a square ABCD of side 10cm, as shown in figure. The force on a charge of 1μC placed at the centre of square is….​
17. Five charges q1, q2, q3, q4 and q5 are fixed at their positions as shown in figure. S is Gaussian surface. The Gauss? law is given by ∮s→E⋅→ds=qε0. Which of the following statements is correct?​
18. Five styrofoam balls are suspended from insulating threads. Several experiments are performed on the balls and the following observations are made - (i) Ball A repels C and attracts B (ii) Ball D attracts B and has no effect on E (iii) A negatively charged rod attracts both A and E. An electrically neutral styrofoam ball gets attracted if placed nearby a charged body due to induced charge. What are the charges, if any, on each ball?​
19. Figure below shows a closed surface which intersects a conducting sphere. If a positive charge is placed at the point P, the flux of the electric field through the closed surface​…
20. Figure shows electric field lines in which an electric dipole →p→ is placed as shown. Which of the following statements is correct?

Things to Remember

• Electric dipole is a pair of electric charges possessing equal magnitude but opposite charges separated by distance.
• In simple words, an electric dipole is a separation between positive and negative charges.
• It is worthy to note that torque always operates in a couple.
• Additionally, its magnitude is equal to the resultant product of force and its arm.
• The torque is the cross product of the dipole moment and electric field.
• The torque is in the clockwise direction (hence negative) in the above figure if the direction of the Electric field is positive.